What do I do when I know that a person is manipulating the financial statements of a private company in order to smooth income throughout the year?

It depends in what capacity you identify the manipulation. for e.g, if you are an external auditor, you can identify the manipulation and suggest an adjusting entry to correct it. if adjustment entry is accepted and passed you do not have to do anything else. otherwise, on the basis of materiality, you can give a qualified opinion.

The treatement of financial statement manipulation usually depends on how material that manipulation and what effects will have on the presentation of the financial statements. 

Since income smoothing will not affect the year end income statement.

The  first thing u can do u can report to those who charged with the governance.other wise you are in charged with that position you can reverse the amount what he/she has wrongly posted.

internal control system must have Procedures prevent fraud but in this you must do report to high management